Smaller Elements of Pythagorean Triple not both Odd

Theorem

Let $\left({x, y, z}\right)$ be a Pythagorean triple, i.e. integers such that $x^2 + y^2 = z^2$.

Then $x$ and $y$ cannot both be odd.

Proof

Aiming for a contradiction, suppose $x$ and $y$ are both odd such that:

$\exists z \in \Z: x^2 + y^2 = z^2$

Then:

$x^2 + y^2 \equiv 1 + 1 \equiv 2 \pmod 4$

But from Square Modulo 4:

$z^2 \equiv 0 \pmod 4$ or $z^2 \equiv 1 \pmod 4$

Thus $x^2 + y^2$ can not be square.

It follows by Proof by Contradiction that $x$ and $y$ cannot both be odd.

$\blacksquare$